ON THE MAXIMAL SPECTRAL TYPE OF NILSYSTEMS

Ethan Ackelsberg; Florian K. Richter; O. R. Shalom

doi:10.1090/bproc/229

ON THE MAXIMAL SPECTRAL TYPE OF NILSYSTEMS

Ethan Ackelsberg, Florian K. Richter, O. R. Shalom

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Abstract

Let (G/Γ, R_a) be an ergodic k-step nilsystem for k ≥ 2. We adapt an argument of Parry [Topology 9 (1970), pp. 217–224] to show that L² (G/Γ) decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host–Kra–Maass [J. Anal. Math. 124 (2014), pp. 261–295] for 2-step nilsystems and a result by Stepin [Uspehi Mat. Nauk 24 (1969), pp. 241–242] for nilsystems G/Γ with connected, simply connected G.

Original language	English
Pages (from-to)	469-480
Number of pages	12
Journal	Proceedings of the American Mathematical Society, Series B
Volume	11
Issue number	1
DOIs	https://doi.org/10.1090/bproc/229
State	Published - 2024
Externally published	Yes

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abstract = "Let (G/Γ, Ra) be an ergodic k-step nilsystem for k ≥ 2. We adapt an argument of Parry [Topology 9 (1970), pp. 217–224] to show that L2 (G/Γ) decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host–Kra–Maass [J. Anal. Math. 124 (2014), pp. 261–295] for 2-step nilsystems and a result by Stepin [Uspehi Mat. Nauk 24 (1969), pp. 241–242] for nilsystems G/Γ with connected, simply connected G.",

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AU - Shalom, O. R.

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N2 - Let (G/Γ, Ra) be an ergodic k-step nilsystem for k ≥ 2. We adapt an argument of Parry [Topology 9 (1970), pp. 217–224] to show that L2 (G/Γ) decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host–Kra–Maass [J. Anal. Math. 124 (2014), pp. 261–295] for 2-step nilsystems and a result by Stepin [Uspehi Mat. Nauk 24 (1969), pp. 241–242] for nilsystems G/Γ with connected, simply connected G.

AB - Let (G/Γ, Ra) be an ergodic k-step nilsystem for k ≥ 2. We adapt an argument of Parry [Topology 9 (1970), pp. 217–224] to show that L2 (G/Γ) decomposes as a sum of a subspace with discrete spectrum and a subspace of Lebesgue spectrum with infinite multiplicity. In particular, we generalize a result previously established by Host–Kra–Maass [J. Anal. Math. 124 (2014), pp. 261–295] for 2-step nilsystems and a result by Stepin [Uspehi Mat. Nauk 24 (1969), pp. 241–242] for nilsystems G/Γ with connected, simply connected G.

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ON THE MAXIMAL SPECTRAL TYPE OF NILSYSTEMS

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